466 Higher Engineering Mathematics
Table 49.5
x y
0 1
0.2 1
0.4 0.96
0.6 0.8864
0.8 0.793664
1.0 0.699692
(b) If the solution of the differential equa-
tion by an analytical method is given
byy=
4
x
−
x
2
, determine the percentage
error atx= 2. 2
[(a) see Table 49.6 (b) 1.206%]
Table 49.6
x y
2.0 1
2.1 0.85
2.2 0.709524
2.3 0.577273
2.4 0.452174
2.5 0.333334
- UseEuler’smethodtoobtainanumerical solu-
tion of the differential equation
dy
dx
=x−
2 y
x
,
given the initial conditions thaty=1when
x=2, in the rangex= 2. 0 ( 0. 2 ) 3 .0.
If the solution of the differential equation is
given byy=
x^2
4
, determine the percentage
error by using Euler’s method whenx= 2. 8
[see Table 49.7, 1.596%]
Table 49.7
x y
2.0 1
2.2 1.2
2.4 1.421818
2.6 1.664849
2.8 1.928718
3.0 2.213187
49.4 An improved Euler method
In Euler’s method of Section 49, the gradient(y′) 0 at
P(x 0 ,y 0 )in Fig. 49.9 across the whole intervalhis used
to obtain an approximate value ofy 1 at pointQ.QRin
Fig. 49.9 is the resulting error in the result.
y
P
x 0 x 1 x
Q
R
y 0
0
h
Figure 49.9
In an improved Euler method, called theEuler-Cauchy
method, the gradient atP(x 0 ,y 0 )across half the interval
is used and then continues with a line whose gradient
approximates to the gradient of the curve atx 1 ,shown
in Fig. 49.10.
LetyP 1 be the predicted value at pointRusing Euler’s
method, i.e. lengthRZ,where
yP 1 =y 0 +h(y′) 0 (3)
The error shown asQTin Fig. 49.10 is now less
than the errorQRused in the basic Euler method and
the calculated results will be of greater accuracy. The
y
P S
x 0 x 1 x
Q
R
Z
T
0
h
x 0112 h
Figure 49.10